1. Field of the Invention
This invention pertains in general to the field of phase-shifting interferometry and, in particular, to a novel approach for mapping the profile of the surface of a test sample taking into account phase shifts related to the optical parameters of the sample material.
2. Description of the Related Art
Optical surface profilers based on phase-shifting interferometry (PSI) utilize phase measurements to calculate the surface height values, h(x,y), at each point of a surface under test. The technique is founded on the concept of varying the phase difference between two coherent interfering beams of single wavelength in some known manner, such as by changing the optical path difference (OPD) in discrete steps or linearly with time. Under such conditions, three or more measurements of the light intensity at a pixel of a receiving sensor array can be used to determine the initial phase difference of the light beams at the point on the test surface corresponding to that pixel. Based on such measurements at each pixel of coordinates x and y, a phase distribution map .phi.(x,y) can be determined for the test surface, from which very accurate height data h(x,y) are calculated by the following general equation in relation to the wavelength .lambda. of the light source used: ##EQU1## Phase-shifting interferometry provides a vertical resolution of the order of 1/100 of a wavelength or better and is widely used for measuring opaque surfaces of similar (homogeneous) materials. If the sample material is not dielectric (i.e., the extinction coefficient of the material is not zero), a phase change occurs on reflection from the sample (referred to in the art as Fresnel phase change on reflections). The phase changes on reflection from the surface of the sample vary with several parameters, including the optical constants of the material composing the surface of the sample. However, existing techniques for reconstructing surface profiles from phase measurements do not take into account phase changes that result from reflection of the light incident on the surface of the sample. This approximation is not a problem when the test sample's surface is made of similar material because the phase shift due to the optical parameters of the material is the same at each pixel. Since only relative phase changes from pixel to pixel are important for determining a surface profile, phase changes related to optical parameters can be neglected under these circumstances. On the other hand, when the sample surface is dissimilar (non-homogeneous), the optical-parameter variations from pixel to pixel result in nonuniform phase shifts that distort the measured surface profile. Therefore, unless these parameters are also known and accounted for, a correct profile measurement is improbable with conventional techniques.
The relationship between the phase change associated with a beam reflected at the interface between an incident medium (such as air) and a sample surface and the physical properties of the sample material is well understood in the art. For example, referring to the general case where a plane wave of monochromatic, linearly-polarized light of wavelength .lambda. is incident on the surface of an optically-opaque sample at an incidence angle .theta..sub.i, the phase change .DELTA..phi. of the beam reflected from the surface can be calculated by known equations having the following general functionality: EQU .DELTA..phi..sub.TE f.sub.TE (n,k,.theta..sub.i,.lambda.) (2a) EQU .DELTA..phi..sub.TM f.sub.TM (n,k,.theta..sub.i,.lambda.) (2b)
where TE and TM refer to the axes of polarization of the incident light, TE being parallel and TM being perpendicular, respectively, to the plane of incidence; and n and k are the refractive index and the extinction coefficient of the sample material, respectively. For details of the specific equations used in the art to define the relationship between these variables, see Born, Max and Emil Wolf, "Principles of Optics," 4th Edition, Pergamon Press, Bath, England, at pp. 615 and sequel.
It is known that the optical parameters n and k vary with the wavelength .lambda. of the incident light; therefore, for given wavelength and angle of incidence, Equations 2a and 2b can be expressed simply in terms of n an k; that, is, .DELTA..phi.=.DELTA..phi.(n,k). Accordingly, the phase change .DELTA..phi. of incident light of wavelength .lambda. reflected from a sample surface can be calculated exactly for each pixel if the refractive index and extinction coefficient at that wavelength and the angle of incidence are known for that pixel. Once a map of such phase shifts .DELTA..phi.(x,Y) is known, a corrected phase distribution map .phi..sub.CORR (x,Y) can be determined by EQU .phi..sub.CORR (x,y)=.phi.(x,y)+.DELTA..phi.(x,Y) (3)
and used to calculate a corrected height map using Equation 1, which becomes ##EQU2##
Prior-art phase shifting techniques have completely neglected this correction because n and k are normally unknown for the material being tested. Also, as mentioned above, this correction is unnecessary for surfaces of similar material.
The quantities n and k are conventionally measured by ellipsometric techniques and vary with the wavelength of the light used for testing. In addition, they are usually not uniform within the surface of the test sample. Therefore, the refractive index and the extinction coefficient of the sample material are point quantities that in practice are not available during phase-shifting measurements for correction of errors introduced by nonuniformities or dissimilarities within the surface of the test sample. This aspect of phase-shifting interferometry remains a problem in the continuing effort to improve the accuracy and resolution of the technique.
The present invention is directed at providing an approach that improves prior-art techniques by estimating n and k at each measurement pixel of the material being tested and by obtaining phase-shift measurements that account for the phase changes related to the optical parameters of the sample surface.